The present invention relates to a viscosity modifier for lubricating oil and a lubricating oil composition. More particularly, the invention relates to a viscosity modifier for lubricating oil, by the use of which a lubricating oil composition having excellent low-temperature properties and lubricity at high temperatures can be obtained, and a lubricating oil composition containing the viscosity modifier.
The viscosity of petroleum products generally varies greatly with temperature change, but the temperature dependence of the viscosity of, for example, a lubricating oil for automobile is desired to be small. On this account, an ethylene/xcex1-olefin copolymer having a viscosity index improving effect has been widely used for the lubricating oil to make the temperature dependence smaller.
The lubricating oil loses flowability at low temperatures because a wax component in the lubricating oil is crystallized and solidified at low temperatures. In order to lower the solidifying temperature, a pour point depressant is contained in the lubricating oil, and the pour point depressant hinders formation of three-dimensional network that is due to crystallization of the wax component in the lubricating oil, and thereby lowers the pour point of the lubricating oil.
Of the low-temperature properties of the lubricating oil containing an ethylene/xcex1-olefin copolymer and a pour point depressant, the viscosity under the conditions of a high shear rate depends upon compatibility of a lubricating oil base with the ethylene/xcex1-olefin copolymer, while the viscosity under the conditions of a low shear rate is greatly influenced by the pour point depressant. It is known that when an ethylene/xcex1-olefin copolymer having a specific composition is used, the effect of the pour point depressant is markedly reduced by the interaction between the copolymer and the pour point depressant (see U.S. Pat. No. 3,697,429 and U.S. Pat. No. 3,551,336).
Therefore, the ethylene/xcex1-olefin copolymer added to the lubricating oil that particularly needs excellent low-temperature properties is desired not only to have an excellent viscosity index improving effect but also not to hinder the function of the pour point depressant.
In order to prevent the interaction between the pour point depressant and the ethylene/xcex1-olefin copolymer, use of an ethylene/xcex1-olefin copolymer obtained by a specific polymerizing apparatus and conditions and having an ununiform composition distribution as a viscosity index improver has been proposed (see Japanese Patent Laid-Open Publication No. 228600/1985). However, any lubricating oil having excellent low-temperature properties independent of the shear rate has not been obtained so far.
Under such circumstances as described above, the present inventor has made earnest studies and has found that by the use of a specific ethylene/xcex1-olefin copolymer, decrease of the effect of the pour point depressant caused by the aforesaid interaction is not brought about, the compatibility with the lubricating oil base at low temperatures is adjusted, and thereby a lubricating oil having excellent low-temperature properties in every shear rate region can be obtained. Based on the finding, the present invention has been accomplished.
It is an object of the present invention to provide a lubricating oil composition that is excellent in low-temperature properties, oxidative stability, lubricity at high temperatures and saving of fuel.
The viscosity modifier for lubricating oil according to the invention comprises an ethylene/xcex1-olefin copolymer which is a copolymer of ethylene and an xcex1-olefin of 3 to 20 carbon atoms and has the following properties:
(1) the ethylene content (E) is in the range of 40 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than 60xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the intensity ratio, Sxcex1xcex2/Sxcex1xcex1, measured by 13C-NMR spectrum is not more than 0.5.
The ethylene/xcex1-olefin copolymer is preferably an ethylene/propylene copolymer.
The lubricating oil composition according to the invention is a composition comprising:
(A) a lubricating oil base, and
(B) an ethylene/xcex1-olefin copolymer which is a copolymer of ethylene and an o-olefin of 3 to 20 carbon atoms and has the following properties:
(1) the ethylene content (E) is in the range of 40 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than 60xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the intensity ratio, Sxcex1xcex2/Sxcex1xcex1, measured by 13C-NMR spectrum is not more than 0.5;
wherein the ethylene/xcex1-olefin polymer (B) is contained in an amount of 1 to 20% by weight.
Further, the lubricating oil composition according to the invention is a composition comprising the lubricating oil base (A), the ethylene/xcex1-olefin copolymer (B) and a pour point depressant (C), wherein:
the ethylene/xcex1-olefin copolymer (B) is contained in an amount of 0.1 to 5% by weight, and
the pour point depressant (C) is contained in an amount of 0.05 to 5% by weight.
The ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer (B-1) having the following properties:
(1) the ethylene content (E) is in the range of 65 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is in the range of 15 to 60xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the intensity ratio, Sxcex1xcex2/Sxcex1xcex1, measured by 13C-NMR spectrum is not more than 0.5.
Further, the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer (B-2) having the following properties:
(1) the ethylene content (E) is in the range of 40 to 55% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than xe2x88x9220xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the intensity ratio, Sxcex1xcex2/Sxcex1xcex1, measured by 13C-NMR spectrum is not more than 0.5.
In the lubricating oil composition, the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/propylene copolymer.
The ethylene/xcex1-olefin copolymer (B) is preferably a copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg ranging from 8 to 16.
Further, the ethylene/xcex1-olefin copolymer (B) is preferably a copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg being not less than 5 and less than 8.
The viscosity modifier for lubricating oil and the lubricating oil composition according to the invention are described in detail hereinafter.
The viscosity modifier for lubricating oil according to the invention comprises a copolymer of ethylene and an xcex1-olefin (sometimes referred to simply as an xe2x80x9cethylene/xcex1-olefin copolymerxe2x80x9d simply hereinafter in this specification) having the following properties.
The ethylene/xcex1-olefin copolymer has an ethylene content (E) (content of recurring units derived from ethylene) of usually 40 to 77% by weight. The ethylene content (E) is desired to be in the range of 65 to 77% by weight, preferably 68 to 74% by weight, particularly preferably 68 to 72% by weight, or in the range of 40 to 55% by weight, preferably 43 to 53% by weight, particularly preferably 45 to 51% by weight.
When the ethylene content is in the above range, a lubricating oil composition having satisfactory low-temperature properties can be obtained, and besides jellying of the lubricating oil composition at low temperatures due to crystallization of the ethylene sequence portions of the ethylene/xcex1-olefin copolymer does not take place.
The ethylene content (E) in the ethylene/xcex1-olefin copolymer can be determined by 13C-NMR in accordance with the method described in xe2x80x9cMacromolecule Analysis Handbookxe2x80x9d(Society of Japan Analytical Chemistry, edited by Macromolecule Analytical Research Meeting, published by Kinokuniya Shoten).
Examples of the xcex1-olefins of 3 to 20 carbon atoms include propylene, 1-butene, 1-pentene, 1-hexene, 3-methyl-1-butene, 3-methyl-1-pentene, 3-ethyl-1-pentene, 4-methyl-1-pentene, 4-methyl-1-hexene, 4,4-dimethyl-1-hexene, 4,4-dimethyl-1-pentene, 4-ethyl-1-hexene, 3-ethyl-1-hexene, 1-octene, 1-decene, 1-dodecene, 1-tetradecene, 1-hexadecene, 1-octadecene and 1-eicosene.
Of the xcex1-olefins, propylene is particularly preferable. That is, the ethylene/xcex1-olefin copolymer is preferably an ethylene/propylene copolymer.
The ethylene/xcex1-olefin copolymer for use in the invention preferably substantially consists of only constituent units derived from ethylene and constituent units derived from an xcex1-olefin. The ethylene/xcex1-olefin copolymer of this constitution is excellent particularly in low-temperature properties.
Moreover, when the ethylene/xcex1-olefin copolymer does not contain a polyene and the like, it is excellent in heat resistance, especially free of oxidization and coloring, and excellent particularly in lubricating properties at high temperature.
The ethylene/xcex1-olefin copolymer has a molecular weight, as measured in terms of a weight-average molecular weight (Mw) of polystyrene by gel permeation chromatography (GPC), of 80,000 to 400,000. When the weight-average molecular weight (Mw) is in the above range, the ethylene/xcex1-olefin copolymer has excellent viscosity index improvability (thickening ability), and hence a small amount of the ethylene/xcex1-olefin copolymer suffices to obtain a specific lubricating oil viscosity. Moreover, jellying hardly takes place at low temperatures, and the shear stability is good.
When the weight-average molecular weight (Mw) of the ethylene/xcex1-olefin copolymer is in the range of 260,000 to 380,000, preferably 270,000 to 350,000, the viscosity index of the lubricating oil can be particularly improved. When the weight-average molecular weight (Mw) thereof is in the range of 100,000 to 240,000, preferably 120,000 to 240,000, the shear stability of the lubricating oil viscosity can be particularly improved.
The measurement by gel permeation chromatography (GPC) is carried out under the conditions of a temperature of 140xc2x0 C. and a solvent of orthodichlorobenzene.
The ethylene/xcex1-olefin copolymer desirably has Mw/Mn (Mw: weight-average molecular weight, Mn: number-average molecular weight), that is an indication of a molecular weight distribution of the copolymer, of not more than 2.4, preferably not more than 2.2. If the molecular weight distribution is not more than 2.4, the shear stability of the lubricating oil viscosity is good.
The ethylene/xcex1-olefin copolymer has a melting point (Tm), as measured by a differential scanning calorimeter (DSC), of not higher than 60xc2x0 C.
In particular, the melting point (Tm) of the ethylene/xcex1-olefin copolymer having an ethylene content (E) of 65 to 77% by weight is desired to be in the range of 15 to 60xc2x0 C., preferably 25 to 50xc2x0 C., more preferably 25 to 45xc2x0 C. The melting point (Tm) of the ethylene/xcex1-olefin copolymer having an ethylene content (E) of 40 to 55% by weight, as measured by a differential scanning calorimeter (DSC), is desired to be not higher than xe2x88x9220xc2x0 C., preferably not higher than xe2x88x9225xc2x0 C., more preferably not higher than xe2x88x9230xc2x0 C.
The melting point is determined by obtaining an endothermic curve by means of a differential scanning calorimeter (DSC), and the temperature at the maximum peak position of the endothermic curve is taken as the melting point. More specifically, a sample is placed in an aluminum pan, heated up to 200xc2x0 C. at a rate of 10xc2x0 C./min, maintained at 200xc2x0 C. for 5 minutes, cooled to xe2x88x92150xc2x0 C. at a rate of 20xc2x0 C./min and then heated at a rate of 10xc2x0 C./min to obtain a 2nd run endothermic curve, from which the melting point is determined.
The ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.) as measured by DSC preferably satisfy the following relation (relational expression)
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I) 
The relational expression (I) is a measure of composition distribution, and if the relationship between the ethylene content and the melting point is out of the relational expression (I), the composition distribution of the ethylene/xcex1-olefin copolymer is widened to thereby lower the low-temperature resistance of the lubricating oil, and besides a problem of opaqueness (haze) of the lubricating oil is brought about by the presence of a high ethylene content portion. In the present invention, the ethylene/xcex1-olefin copolymer is preferably one obtained by copolymerizing ethylene and an xcex1-olefin using a catalyst comprising the later-described metallocene compound and ionizing ionic compound from the viewpoint of composition distribution.
The relational expression (I) as the composition distribution parameter is:
preferably
3.31xc3x97Exe2x88x92192xe2x89xa7Tmxe2x80x83xe2x80x83(Ixe2x80x2) 
particularly preferably
3.31xc3x97Exe2x88x92192xe2x89xa7Tmxe2x80x83xe2x80x83(Ixe2x80x3) 
Preferably, the ethylene, propylene and xcex1-olefin copolymer of the invention satisfy the intensity ratio, Sxcex1xcex2/Sxcex1xcex1, measured by 13C-NMR spectrum of not more than 0.5. Each of the Sxcex1xcex2 and Sxcex1xcex1 measured by 13C-NMR spectrum is a peak intensity of CH2 in the units derived from ethylene or an xcex1-olefin of 3 or more carbon atoms and they mean two kinds of CH2 different in the position to the tertiary carbon as shown below; 
The 13C-NMR spectrum thus measured was analyzed and then Sxcex1xcex2, Sxcex1xcex1 are determined in accordance with the method reported by J. C. Randall (Macromolecules, 11, 33 (1978)).
The intensity ratio Sxcex1xcex2/Sxcex1xcex1 is calculated from a ratio of an integral value (area) of each peak. It is generally considered that the thus obtained value of the intensity ratio is a measure indicating a ratio of such reactions as an occurrence of 1-2 addition reaction of xcex1-olefin followed by 2-1 addition reaction or an occurrence of 2-1 addition reaction of xcex1-olefin followed by 1-2 addition reaction. Consequently, it is indicated that the larger the intensity ratio is, more irregular the bonding direction of xcex1-olefin is. On the other hand, the smaller the intensity ratio is, more regular the bonding direction of xcex1-olefin is.
When the ratio satisfies the above formula (Sxcex1xcex2/Sxcex1xcex1xe2x89xa60.5), the fluidity of a lubricating oil at law temperature is improved and the lubricating properties at high temperature can also be improved. Further, the lubricating oil shows excellent balance of the lubricating properties at low temperature and high temperature.
Especially when film strength of an engine oil is required, the ratio (I10/I2) of a melt index of the ethylene/xcex1-olefin copolymer as measured at 190xc2x0 C. under a load of 10 kg to a melt index of the copolymer as measured at 190xc2x0 C. under a load of 2.16 kg is desired to be in the range of 8 to 16, preferably 8.0 to 16, more preferably 8.0 to 16.0, still more preferably 8.0 to 13, particularly preferably 8.0 to 13.0.
Especially when shear stability of an engine oil is required, the ratio (I10/I2) of a melt index of the ethylene/xcex1-olefin copolymer as measured at 190xc2x0 C. under a load of 10 kg to a melt index of the copolymer as measured at 190xc2x0 C. under a load of 2.16 kg is desired to be not less than 5 and less than 8, preferably not less than 5.0 and less than 8.0, more preferably not less than 5.5 and less than 7.5, still more preferably not less than 5.8 and less than 7.0.
The ethylene/xcex1-olefin copolymer exhibits a high viscosity index improving effect and hardly hinder the effect of a pour point depressant when blended with a lubricating oil base.
When the ethylene/xcex1-olefin copolymer is used as a viscosity modifier, a lubricating oil capable of satisfying low-temperature property standards of the GF-3 standards, which are standards of North America Lubricating Oil Standards of the next generation, can be obtained. Whether the lubricating oil satisfies the GF-3 standards or not can be judged by measuring the later-described CCS and MRV.
The ethylene/xcex1-olefin copolymer for use as the lubricating oil viscosity modifier of the invention can be obtained by copolymerizing ethylene, an xcex1-olefin of 3 to 20 carbon atoms, and if necessary, other monomers in the presence of an olefin polymerization catalyst.
Examples of the olefin polymerization catalysts employable herein include catalysts comprising a compound of a transition metal such as zirconium, hafnium or titanium, and an organoaluminum compound (organoaluminum oxy-compound) and/or an ionizing ionic compound. Of these, a metallocene catalyst comprising a metallocene compound of a transition metal selected from Group 4, etc. of the periodic table and an organoaluminum oxy-compound and/or an ionizing ionic compound is particularly preferably used in the invention.
Next, the metallocene catalyst is described.
The metallocene compound of a transition metal of Group 4 of the periodic table for forming the metallocene catalyst is specifically represented by the following formula (i).
MLxxe2x80x83xe2x80x83(i) 
In the formula (i), M is a transition metal selected from Group 4 of the periodic table, specifically zirconium, titanium or hafnium, and x is a valence of the transition metal.
L is a ligand coordinated to the transition metal, and of the ligands, at least one ligand L is a ligand having cyclopentadienyl skeleton. The ligand having cyclopentadienyl skeleton may have a substituent group.
Examples of the ligands having cyclopentadienyl skeleton include a cyclopentadienyl group; and alkyl-substituted or cycloalkyl-substituted cyclopentadienyl groups, such as methylcyclopentadienyl, ethylcyclopentadienyl, n- or i-propylcyclopentadienyl, n-, i-, sec- or t-butylcyclopentadienyl, hexylcyclopentadienyl, octylcyclopentadienyl, dimethylcyclopentadienyl, trimethylcyclopentadienyl, tetramethylcyclopentadienyl, pentamethylcyclopentadienyl, methylethylcyclopentadienyl, methylpropylcyclopentadienyl, methylbutylcyclopentadienyl, methylhexylcyclopentadienyl, methylbenzylcyclopentadienyl, ethylbutylcyclopentadienyl, ethylhexylcyclopentadienyl and methylcyclohexylcyclopentadienyl; an indenyl group; a 4,5,6,7-tetrahydroindenyl group; and a fluorenyl group.
These groups may be substituted with halogen atoms and trialkylsilyl groups.
Of the above groups, alkyl-substituted cyclopentadienyl groups are particularly preferable.
When the compound represented by the formula (i) contains as the ligands L two or more groups having cyclopentadienyl skeleton, two of them may be bonded through an alkylene group such as ethylene or propylene, a substituted alkylene group such as isopropylidene or diphenylmethylene, a silylene group, or a substituted silylene group such as dimethylsilylene, diphenylsilylene or methylphenylsilylene.
The ligand L other than the ligand having cyclopentadienyl skeleton is, for example, a hydrocarbon group of 1 to 12 carbon atoms, an alkoxy group, an aryloxy group, a sulfonic acid-containing group (xe2x80x94SO3Ra wherein Ra is an alkyl group, a halogen-substituted alkyl group, an aryl group, a halogen-substituted aryl group or an alkyl-substituted aryl group), a halogen atom or a hydrogen atom.
Examples of the hydrocarbon groups of 1 to 12 carbon atoms include alkyl groups, cycloalkyl groups, aryl groups and aralkyl groups. More specifically, there can be mentioned:
alkyl groups, such as methyl, ethyl, n-propyl, isopropyl, n-butyl, isobutyl, sec-butyl, t-butyl, pentyl, hexyl, octyl, decyl and dodecyl;
cycloalkyl groups, such as cyclopentyl and cyclohexyl;
aryl groups, such as phenyl and tolyl; and
aralkyl groups, such as benzyl and neophyl.
Examples of the alkoxy groups include methoxy, ethoxy, n-propoxy, isopropoxy, n-butoxy, isobutoxy, sec-butoxy, t-butoxy, pentoxy, hexoxy and octoxy.
An example of the aryloxy group is phenoxy.
Examples of the sulfonic acid-containing groups (xe2x80x94SO3Ra) include methanesulfonato, p-toluenesulfonato, trifluoromethanesulfonato and p-chlorobenzenesulfonato.
The halogen atom is fluorine, chlorine, bromine or iodine.
Examples of the metallocene compounds having zirconium as M and containing at least two ligands having cyclopentadienyl skeleton include:
bis(methylcyclopentadienyl)zirconium dichloride,
bis(ethylcyclopentadienyl)zirconium dichloride,
bis(n-propylcyclopentadienyl)zirconium dichloride,
bis(indenyl)zirconium dichloride, and
bis(4,5,6,7-tetrahydroindenyl)zirconium dichloride.
Compounds wherein the zirconium metal is replaced with a titanium metal or a hafnium metal in the above compounds are also employable.
Also employable as the metallocene compound is a compound represented by the following formula (ii):
L1M1X2xe2x80x83xe2x80x83(ii) 
wherein M is a metal of Group 4 of the periodic table or a metal of lanthanide series,
L1 is a derivative of a delocalized n-bonding group and imparts a restraint geometric shape to the active site of the metal M1, and
each X is independently hydrogen, halogen, a hydrocarbon group containing 20 or less carbon atoms, a silyl group containing 20 or less silicon atoms, or a germyl group containing 20 or less germanium atoms.
Of the compounds represented by the formula (ii), a compound represented by the following formula (iii) is preferable. 
In the above formula, M1 is titanium, zirconium or hafnium, and X is the same as above.
Cp is a substituted cyclopentadienyl group xcfx80-bonded to M1 and having a substituent group Z.
Z is oxygen, sulfur, boron or an element of Group 14 of the periodic table (e.g., silicon, germanium or tin).
Y is a ligand containing nitrogen, phosphorus, oxygen or sulfur.
Z and Y may together form a condensed ring.
Examples of the metallocene compounds represented by the formula (iii) include:
(dimethyl(t-butylamido)(tetramethyl-xcex75-cyclopentadienyl)silane)titanium dichloride,
((t-butylamido)(tetramethyl-xcex75-cyclopentadienyl)-1,2-ethanediyl)titanium dichloride,
(dibenzyl(t-butylamido)(tetramethyl-xcex75-cyclopentadienyl)silane)titanium dichloride,
(dimethyl(t-butylamido)(tetramethyl-xcex75-cyclopentadienyl)silane)dibenzyltitanium,
(dimethyl(t-butylamido)(tetramethyl-xcex75-cyclopentadienyl)silane)dimethyltitanium,
((t-butylamido)(tetramethyl-xcex75-cyclopentadienyl)-1,2-ethanediyl)dibenzyltitanium,
((methylamido)(tetramethyl-xcex75-cyclopentadienyl)-1,2-ethanediyl)dineopentyltitanium,
((phenylphosphido)(tetramethyl-xcex75-cyclopentadienyl)methylene)diphenyltitanium,
(dibenzyl(t-butylamido)(tetramethyl-xcex75-cyclopentadienyl)silane)dibenzyltitanium,
(dimethyl(benzylamido)(xcex75-cyclopentadienyl)silane)di(trimethylsilyl)titanium,
(dimethyl(phenylphosphido)(tetramethyl-xcex75-cyclopentadienyl)silane)dibenzyltitanium,
((tetramethyl-xcex75-cyclopentadienyl)-1,2-ethanediyl)dibenzyltitanium,
(2-xcex75-(tetramethyl-cyclopentadienyl)-1-methyl-ethanolate(2-))dibenzyltitanium,
(2-xcex75-(tetramethyl-cyclopentadienyl)-1-methyl-ethanolate(2-))dimethyltitanium,
(2-((4a, 4b, 8a, 9,9a-xcex7)-9H-fluorene-9-yl)cyclohexanolate(2-))dimethyltitanium, and
(2-((4a, 4b, 8a, 9,9a-xcex7)-9H-fluorene-9-yl)cyclohexanolate(2-))dibenzyltitanium.
Compounds wherein the titanium metal is replaced with a zirconium metal or a hafnium metal in the above compounds are also employable.
The metallocene compounds mentioned above can be used singly or in combination of two or more kinds.
In the present invention, a zirconocene compound having zirconium as the central metal atom and containing at least two ligands having cyclopentadienyl skeleton is preferably used as the metallocene compound represented by the formula (i). As the metallocene compound represented by the formula (ii) or (iii), a compound having titanium as the central metal atom is preferable. Of the metallocene compounds mentioned above, a compound represented by the formula (iii) and having titanium as the central metal atom is particularly preferable.
The organoaluminum oxy-compound for forming the metallocene catalyst may be aluminoxane hitherto known or a benzene-insoluble organoaluminum oxy-compound.
The hitherto known aluminoxane is represented by the following formula: 
In the formulas (iv) and (v), R is a hydrocarbon group, such as methyl, ethyl, propyl or butyl, preferably methyl or ethyl, particularly preferably methyl; and m is an integer of 2 or more, preferably an integer of 5 to 40.
The aluminoxane may be composed of mixed alkyloxyaluminum units consisting of alkyloxyaluminum units represented by the formula OAl (R1) and alkyloxyaluminum units represented by the formula OAl (R2) (each of R1 and R2 is the same hydrocarbon group as indicated by R, and R1 and R2 are groups different from each other).
Examples of the ionizing ionic compounds for forming the metallocene catalyst include Lewis acid and an ionic compound.
The Lewis acid is, for example, a compound represented by BR3 (R is a phenyl group which may have a substituent group such as fluorine, methyl or trifluoromethyl, or fluorine). Examples of such compounds include trifluoroboron, triphenylboron, tris(4-fluorophenyl)boron, tris(3,5-difluorophenyl)boron, tris(4-fluoromethylphenyl)boron, tris(pentafluorophenyl)boron, tris(p-tolyl)boron, tris(o-tolyl)boron and tris(3,5-dimethylphenyl)boron.
The ionic compound is, for example, a trialkyl-substituted ammonium salt, an N,N-dialkylanilinium salt, a dialkylammonium salt or a triarylphosphonium salt.
Examples of the trialkyl-substituted ammonium salts include:
triethylammoniumtetra(phenyl)boron,
tripropylammoniumtetra(phenyl)boron,
tri(n-butyl)ammoniumtetra(phenyl)boron,
trimethylammoniumtetra(p-tolyl)boron,
trimethylammoniumtetra(o-tolyl)boron,
tributylammoniumtetra(pentafluorophenyl)boron,
tripropylammoniumtetra(o,p-dimethylphenyl)boron,
tributylammoniumtetra(m,m-dimethylphenyl)boron,
tributylammoniumtetra(p-trifluoromethylphenyl)boron,
and
tri(n-butyl)ammoniumtetra(o-tolyl)boron.
Examples of the N,N-dialkylanilinium salts include:
N,N-dimethylaniliniumtetra(phenyl)boron,
N,N-diethylaniliniumtetra(phenyl)boron, and
N,N-2,4,6-pentamethylaniliniumtetra(phenyl)boron.
Examples of the dialkylammonium salts include:
di(1-propyl)ammoniumtetra(pentafluorophenyl)boron,
and
dicyclohexylammoniumtetra(phenyl)boron.
Also employable as the ionic compound is triphenylcarbeniumtetrakis(pentafluorophenyl)borate, N,N-dimethylaniliniumtetrakis(pentafluorophenyl)borate or ferroceniumtetra(pentafluorophenyl)borate. In particular, the ionizing ionic compound is preferably used because it controls composition distribution of the ethylene/xcex1-olefin copolymer.
For forming the metallocene catalyst, an organoaluminum compound may be used together with the organoaluminum oxy-compound and/or the ionizing ionic compound.
The organoaluminum compound is, for example, a compound represented by the following formula (vi):
R1nAlX3-n xe2x80x83xe2x80x83(vi) 
wherein R1 is a hydrocarbon group of 1 to 15 carbon atoms, preferably 1 to 4 carbon atoms, X is a halogen atom or a hydrogen atom, and n is 1 to 3.
The hydrocarbon group of 1 to 15 carbon atoms is, for example, an alkyl group, a cycloalkyl group or an aryl group. Examples of such groups include methyl, ethyl, n-propyl, isopropyl, isobutyl, pentyl, hexyl, octyl, cyclopentyl, cyclohexyl, phenyl and tolyl.
Examples of the organoaluminum compounds include:
trialkylaluminums, such as trimethylaluminum, triethylaluminum, triisopropylaluminum, triisobutylaluminum, trioctylaluminum and tri-2-ethylhexylaluminum;
alkenylaluminums represented by the formula (i-C4H9)xAly (C5H10)z (wherein x, y and z are each a positive number, and zxe2x89xa72x), such as isoprenylaluminum;
trialkenylaluminums, such as triisopropenylaluminum;
dialkylaluminum halides, such as dimethylaluminum chloride, diethylaluminum chloride, diisopropylaluminum chloride, diisobutylaluminum chloride and dimethylaluminum bromide;
alkylaluminum sesquihalides, such as methylaluminum sesquichloride, ethylaluminum sesquichloride, isopropylaluminum sesquichloride, butylaluminum sesquichloride and ethylaluminum sesquibromide,
alkylaluminum dihalides, such as methylaluminum dichloride, ethylaluminum dichloride, isopropylaluminum dichloride and ethylaluminum dibromide;
dialkylaluminum hydrides, such as diethylaluminum hydride and dibutylaluminum hydride; and
alkylaluminum dihydrides, such as ethylaluminum dihydride and propylaluminum dihydride.
In the present invention, ethylene, an xcex1-olefin of 3 to 20 carbon atoms, and if necessary, other monomers are copolymerized in the presence of the metallocene catalyst, usually in a liquid phase. In the copolymerization, a hydrocarbon solvent is generally used as a polymerization solvent, but an xcex1-olefin such as propylene may be used as the solvent.
Examples of the hydrocarbon solvents used in the polymerization include aliphatic hydrocarbons, such as pentane, hexane, heptane, octane, decane, dodecane and kerosine, and halogen derivatives thereof; alicyclic hydrocarbons, such as cyclohexane, methylcyclopentane and methylcyclohexane, and halogen derivatives thereof; and aromatic hydrocarbons, such as benzene, toluene and xylene, and halogen derivatives thereof such as chlorobenzene. These solvents can be used singly or in combination of two or more kinds.
Although ethylene, an xcex1-olefin of 3 to 20 carbon atoms, and if necessary, other monomers can be copolymerized by any of batchwise and continuous processes, the copolymerization is preferably carried out continuously, and is particularly preferably carried out continuously using a stirring vessel type reactor. When the copolymerization is carried out continuously, the metallocene catalyst is used in, for example, the following concentration.
The concentration of the metallocene compound in the polymerization system is in the range of usually 0.00005 to 0.1 mmol/liter (polymerization volume), preferably 0.0001 to 0.05 mmol/liter. The organoaluminum oxy-compound is fed in such an amount that the molar ratio (Al/transition metal) of the aluminum atom to the transition metal in the metallocene compound in the polymerization system becomes usually 1 to 10000, preferably 10 to 5000.
The ionizing ionic compound is fed in such an amount that the molar ratio (ionizing ionic compound/metallocene compound) of the ionizing ionic compound to the metallocene compound in the polymerization system becomes 0.5 to 30, preferably 1 to 25.
When the organoaluminum compound is used, the compound is fed in an amount of usually about 0 to 5 mmol/liter (polymerization volume), preferably about 0 to 2 mmol/liter.
In the copolymerization of ethylene, an xcex1-olefin of 3 to 20 carbon atoms, and if necessary, other monomers in the presence of the metallocene catalyst, the copolymerization reaction is carried out under the conditions of a temperature of usually xe2x88x9220 to 150xc2x0 C., preferably 0 to 120xc2x0 C., more preferably 0 to 100xc2x0 C., and a pressure of more than 0 kg/cm2 and not more than 80 kg/cm2, preferably more than 0 kg/cm2 and not more than 50 kg/cm2. In a continuous polymerization process, these polymerization conditions are preferably held constant.
Although the reaction time (average residence time when the copolymerization is carried out continuously) varies depending upon the catalyst concentration, polymerization temperature, etc., it is in the range of usually 5 minutes to 5 hours, preferably 10 minutes to 3 hours.
Ethylene, an xcex1-olefin of 3 to 20 carbon atoms, and if necessary, other monomers are fed to the polymerization system in such amounts that the ethylene/xcex1-olefin copolymer having specific composition can be obtained. In the copolymerization, a molecular weight modifier such as hydrogen may be further used.
When ethylene, an xcex1-olefin of 3 to 20 carbon atoms, and if necessary, other monomers are copolymerized as described above, the ethylene/xcex1-olefin copolymer is usually obtained as a polymer solution containing the copolymer. The polymer solution is treated in a conventional manner, whereby the ethylene/xcex1-olefin copolymer for use in the invention is obtained.
The lubricating oil composition according to the invention is a lubricating oil composition comprising a lubricating oil base (A) and the above-mentioned ethylene/xcex1-olefin copolymer (B), or a lubricating oil composition comprising a lubricating oil base (A), the specific ethylene/xcex1-olefin copolymer (B) and a pour point depressant (C).
The components for forming the lubricating oil composition of the invention are described below.
Examples of the lubricating oil bases (A) for use in the invention include mineral oils and synthetic oils such as poly-xcex1-olefins, polyol esters, diesters (e.g., dioctyl phthalate, dioctyl sebacate) and polyalkylene glycols. A mineral oil or a blend of a mineral oil and a synthetic oil is preferably employed. The mineral oil is generally employed after subjected to refining such as dewaxing. Although mineral oils are divided into several classes according to the refining method, a mineral oil having a wax content of 0.5 to 10% is generally employed. For example, a highly refined oil produced by hydrocracking refining, having a low pour point and a high viscosity index and containing isoparaffin as a main component is employable. A mineral oil having a kinematic viscosity of 10 to 200 cSt at 40xc2x0 C. is generally employed.
As the ethylene/xcex1-olefin copolymer (B) for use in the present invention, the aforesaid ethylene/xcex1-olefin copolymer is employed.
In the lubricating oil composition of the invention, particularly an ethylene/xcex1-olefin copolymer (B-1) having the following properties is preferably employed as the ethylene/xcex1-olefin copolymer (B);
(1) the ethylene content (E) is in the range of 65 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is in the range of 15 to 60xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the intensity ratio, Sxcex1xcex2/Sxcex1xcex1, measured by 13C-NMR spectrum is not more than 0.5.
Further, an ethylene/xcex1-olefin copolymer (B-2) having the following properties can be also preferably employed as the ethylene/xcex1-olefin copolymer (B);
(1) the ethylene content (E) is in the range of 40 to 55% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than xe2x88x9220xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the intensity ratio, Sxcex1xcex2/Sxcex1xcex1, measured by 13C-NMR spectrum is not more than 0.5.
The copolymer (B) is preferably an ethylene/xcex1-olefin copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg ranging from 8 to 16.
Further, the copolymer (B) is preferably an ethylene/xcex1-olefin copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg being not less than 5 and less than 8.
Examples of the pour point depressants for use in the invention include alkylated naphthalene, (co)polymers of alkyl methacrylates, (co)polymers of alkyl acrylates, copolymers of alkyl fumarates and vinyl acetate, xcex1-olefin polymers, and copolymers of xcex1-olefins and styrene. Of these, (co)polymers of alkyl methacrylates and (co)polymers of alkyl acrylates are preferably employed.
The first embodiment of the lubricating oil composition of the invention comprises the lubricating oil base (A) and the ethylene/xcex1-olefin copolymer (B). In this lubricating oil composition, the ethylene/xcex1-olefin copolymer (B) is contained in an amount of 1 to 20% by weight, preferably 5 to 10% by weight (residue: lubricating oil base (A) and additives described later).
Such a lubricating oil composition exhibits small temperature dependence and has excellent low-temperature properties. The lubricating oil composition can be used as it is as a lubricating oil, or can be used as a lubricating oil after further blended with a lubricating oil base, a pour depressant, etc.
The second embodiment of the lubricating oil composition of the invention comprises the lubricating oil base (A), the ethylene/xcex1-olefin copolymer (B) and the pour point depressant (C). In this lubricating oil composition, the ethylene/xcex1-olefin copolymer (B) is contained in an amount of 0.1 to 5% by weight, preferably 0.2 to 1.5% by weight, more preferably 0.25 to 1.5% by weight, particularly preferably 0.30 to 1.5% by weight, and the pour point depressant (C) is contained in an amount of 0.05 to 5% by weight, preferably 0.1 to 3% by weight, more preferably 0.1 to 2% by weight, most preferably 0.2 to 1.5% by weight, (residue: lubricating oil base (A) and additives described later). When the amount of the ethylene/xcex1-olefin copolymer (B) in the lubricating oil composition of the invention is not less than 0.1% by weight, a viscosity improving effect can be obtained, and when the amount thereof is not more than 5% by weight, the effect of the pour point depressant (C) is not hindered, so that the above range is preferable. When the amount of the ethylene/xcex1-olefin copolymer (B) is in the above range, a lubricating oil composition having improved viscosity and excellent flowability at low temperatures can be obtained.
Such a lubricating oil composition exhibits small temperature dependence and small rise of the pour point, said rise being attributed to the interaction between the ethylene/xcex1-olefin copolymer and the pour point depressant, and has excellent low-temperature properties in every shear rate region. Moreover, the lubricating oil composition has excellent high-temperature properties and shows good lubricity, so that it is advantageous in saving of fuel.
In the lubricating oil composition of the invention, additives exhibiting viscosity index improving effect, such as (co)polymers of alkyl methacrylates, hydrogenated SBR or SEBS, and other additives, such as detergent, rust preventive agent, dispersant, extreme pressure agent, anti-foaming agent, antioxidant and metal deactivator, may be contained in addition to the lubricating oil base (A), the ethylene/xcex1-olefin copolymer (B) and the pour point depressant (C).
The lubricating oil composition of the invention can be prepared in the following manner: the ethylene/xcex1-olefin copolymer (B), and if necessary, additives are mixed with or dissolved in the lubricating oil base (A) in accordance with a known method; or the ethylene/xcex1-olefin copolymer (B), the pour point depressant (C), and if necessary, additives are mixed with or dissolved in the lubricating oil base (A) in accordance with a known method.
In the present specification, all numerical values for the amounts of materials, reaction conditions, molecular weights, carbon atom numbers, etc. are desirably understood to be supplemented with a term xe2x80x9caboutxe2x80x9d so long as their meanings do not become technically indefinite, with the exception of the following examples or unless otherwise stated.
Further, the following embodiments (i) to (ix) can be mentioned in the present invention.
(i) A viscosity modifier for lubricating oil comprises an ethylene/xcex1-olefin copolymer which is a copolymer of ethylene and an xcex1-olefin of 3 to 20 carbon atoms, and has the following properties:
(1) the ethylene content (E) is in the range of 40 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than 60xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the proportion (V, %) of xcex1xcex2 carbon atoms and the ethylene content (E, % by weight) satisfy the following relation (II):
Vxe2x89xa610xe2x88x920.1xc3x97Exe2x80x83xe2x80x83(II). 
(ii) In the above viscosity modifier for lubricating oil (i), the ethylene/xcex1-olefin copolymer is an ethylene/propylene copolymer.
(iii) A lubricating oil composition comprises:
(A) a lubricating oil base, and
(B) an ethylene/xcex1-olefin copolymer which is a copolymer of ethylene and an xcex1-olefin of 3 to 20 carbon atoms, and has the following properties:
(1) the ethylene content (E) is in the range of 40 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than 60xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the proportion (V, %) of xcex1xcex2 carbon atoms and the ethylene content (E, % by weight) satisfy the following relation:
Vxe2x89xa610xe2x88x920.1xc3x97Exe2x80x83xe2x80x83(II); 
wherein the ethylene/xcex1-olefin polymer (B) is contained in an amount of 1 to 20% by weight.
(iv) A lubricating oil composition comprises:
(A) the lubricating oil base,
(B) the ethylene/xcex1-olefin copolymer, and
(C) a pour point depressant (C),
wherein the ethylene/xcex1-olefin copolymer (B) is contained in an amount of 0.1 to 5% by weight, and the pour point depressant (C) is contained in an amount of 0.05 to 5% by weight.
(v) In the above lubricating oil composition (iii) or (iv), the ethylene/xcex1-olefin copolymer (B) is an ethylene/xcex1-olefin copolymer (B-1) having the following properties:
(1) the ethylene content (E) is in the range of 65 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is in the range of 15 to 60xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the proportion (V, %) of xcex1xcex2 carbon atoms and the ethylene content (E, % by weight) satisfy the following relation (II):
Vxe2x89xa610xe2x88x920.1xc3x97Exe2x80x83xe2x80x83(II). 
(vi) In the above lubricating oil composition (iii) or (iv), the ethylene/xcex1-olefin copolymer (B) is an ethylene/xcex1-olefin copolymer (B-2) having the following properties:
(1) the ethylene content (E) is in the range of 40 to 55% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than xe2x88x9220xc2x0 C.,
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
and
(6) the proportion (V, %) of xcex1xcex2 carbon atoms and the ethylene content (E, % by weight) satisfy the following relation (II):
Vxe2x89xa710xe2x88x920.1xc3x97Exe2x80x83xe2x80x83(II). 
(vii) In any of the above lubricating oil compositions (iii) to (vi), the ethylene/xcex1-olefin copolymer (B) is an ethylene/propylene copolymer.
(viii) In any of the above lubricating oil compositions (iii) to (vii), the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg ranging from 8 to 16.
(ix) In any of the above lubricating oil compositions (iii) to (vii), the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg being not less than 5 and less than 8.
The term xe2x80x9cproportion (V;%) of xcex1xcex2 carbon atomsxe2x80x9d used herein means a proportion of the xcex1xcex2 carbon atoms to specific carbon atoms for forming the ethylene/xcex1-olefin copolymer. The proportion of xcex1xcex2 carbon atoms can be measured by 13C-NMR of the copolymer, followed by determining Sxcex1xcex1, Sxcex1xcex2, Sxcex1xcex4, Sxcex1xcex3, Sxcex1xcex4, Sxcex2xcex4, Sxcex4xcex4, Sxcex3xcex4 and Sxcex2xcex3 in accordance with the method described in J. C. Randall xe2x80x9cMacromoleculesxe2x80x9d(11 33 (1978)). Each of the Sxcex1xcex1, Sxcex1xcex2, Sxcex1xcex4, Sxcex1xcex3, Sxcex1xcex4, Sxcex2xcex4, Sxcex4xcex4, Sxcex3xcex4 and Sxcex2xcex3 measured by 13C-NMR is a peak intensity (integration value) of methylene (CH2) positioned specifically to the tertiary carbon. The proportion is calculated from the following formula.                               Proportion          ⁢                      xe2x80x83                    ⁢          of                                              αβ          ⁢                      xe2x80x83                    ⁢          carbon          ⁢                      xe2x80x83                    ⁢          atoms          ⁢                      xe2x80x83                    ⁢                      (                          V              ;              %                        )                                =                    S        ⁢                  xe2x80x83                ⁢        αβ                                                        (                                                S                  ⁢                                      xe2x80x83                                    ⁢                  αα                                +                                                      3                    /                    4                                    ⁢                  S                  ⁢                                      xe2x80x83                                    ⁢                  αβ                                +                                  S                  ⁢                                      xe2x80x83                                    ⁢                  αγ                                +                                  S                  ⁢                                      xe2x80x83                                    ⁢                  αδ                                +                                                      1                    /                    2                                    ⁢                                      (                                                                  S                        ⁢                                                  xe2x80x83                                                ⁢                        βδ                                            +                                              S                        ⁢                                                  xe2x80x83                                                ⁢                        δδ                                                              )                                                  +                                                      1                    /                    4                                    ⁢                                      (                                                                  S                        ⁢                                                  xe2x80x83                                                ⁢                        γδ                                            +                                              S                        ⁢                                                  xe2x80x83                                                ⁢                        βγ                                                              )                                                                                            xc3x97    100  
If this value (V) satisfies the above relation (Vxe2x89xa610xe2x88x920.1xc3x97E), the fluidity of the lubricating oil at low temperature can be improved and the lubricating properties at high temperature can also be improved. Further, the lubricating oil shows excellent balance of the lubricating properties at low temperature and high temperature.
Still further, the following embodiments (x) to (xviii) can be mentioned in the present invention.
(x) A viscosity modifier for lubricating oil comprises an ethylene/xcex1-olefin copolymer which is a copolymer of ethylene and an xcex1-olefin of 3 to 20 carbon atoms, prepared by using a metallocene catalyst, and has the following properties:
(1) the ethylene content (E) is in the range of 40 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than 60xc2x0 C., and
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I). 
(xi) In the above viscosity modifier for lubricating oil (x), the ethylene/xcex1-olefin copolymer is preferably an ethylene/propylene copolymer.
(xii) A lubricating oil composition comprises:
(A) a lubricating oil base, and
(B) an ethylene/xcex1-olefin copolymer which is a copolymer of ethylene and an xcex1-olefin of 3 to 20 carbon atoms, prepared by using a metallocene catalyst, and has the following properties:
(1) the ethylene content (E) is in the range of 40 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than 60xc2x0 C., and
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I), 
wherein the ethylene/xcex1-olefin copolymer (B) is contained in an amount of 1 to 20% by weight.
(xiii) A lubricating oil composition comprises:
(A) the lubricating oil base,
(B) the ethylene/xcex1-olefin copolymer, and
(C) a pour point depressant,
wherein the ethylene/xcex1-olefin copolymer (B) is contained in an amount of 0.1 to 5% by weight, and the pour point depressant (C) is contained in an amount of 0.05 to 5% by weight.
(xiv) In the above lubricating oil composition (xii) or (xiii), the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer (B-1) prepared by using a metallocene catalyst and having the following properties:
(1) the ethylene content (E) is in the range of 65 to 77% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is in the range of 15 to 60xc2x0 C., and
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I) 
(xv) In the above lubricating oil composition (xii) or (xiii), the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer (B-2) prepared by using a metallocene catalyst and having the following properties:
(1) the ethylene content (E) is in the range of 40 to 55% by weight,
(2) the weight-average molecular weight (Mw) in terms of polystyrene, as measured by GPC, is in the range of 80,000 to 400,000,
(3) Mw/Mn is not more than 2.4,
(4) the melting point (Tm), as measured by DSC, is not higher than xe2x88x9220xc2x0 C., and
(5) the ethylene content (E, % by weight) and the melting point (Tm, xc2x0 C.), as measured by DSC, satisfy the following relation (I):
3.31xc3x97Exe2x88x92186xe2x89xa7Tmxe2x80x83xe2x80x83(I). 
(xvi) In any of the above lubricating oil compositions (xii) to (xv), the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/propylene copolymer.
(xvii) In any of the above lubricating oil compositions (xii) to (xvi), the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg ranging from 8 to 16.
(xviii) In any of the above lubricating oil compositions (xii) to (xvi), the ethylene/xcex1-olefin copolymer (B) is preferably an ethylene/xcex1-olefin copolymer having a ratio (I10/I2) of the melt index (I10) as measured at 190xc2x0 C. under a load of 10 kg to the melt index (I2) as measured at 190xc2x0 C. under a load of 2.16 kg being not less than 5 and less than 8.
The lubricating oil composition of the invention is excellent in the low-temperature properties and the lubricity at high temperatures.